Download Parameter Estimation, Dummy Variables, & Goodness of Fit

Parameter Estimation,
Dummies, & Model Fit
We know mechanically how to “run a
regression”…but how are the
parameters actually estimated?
 How can we handle “categorical”
explanatory (independent)
variables?
 What is a measure of “goodness of
fit” of a statistical model to data?

Example: Alien Species
Exotic species cause economic and
ecological damage
 Not all countries equally invaded
 Want to understand characteristics
of country that make it more likely
to be “invaded”.

Understanding Invasive
Species
Steps to improving our understanding:
1. Generate a set of hypotheses (so they
can be “accepted” or “rejected”)
2. Develop a statistical model. Interpret
hypotheses in context of statistical
model.
3. Collect data. Estimate parameters of
model.
4. Test hypotheses.
2 Hypotheses (in words)
•
We’ll measure “invasiveness” as
proportion of Alien/Native species
(article by Dalmazzone).
1.
Population density plays a role in a
country’s invasiveness.
2.
Island nations are more invaded than
mainland nations.
Population Density
2.0
A.N
1.5
1.0
0.5
0.0
0
200
400
600
Pop.dens
800
1000
1200
Island vs. Mainland
2.0
A.N
1.5
1.0
0.5
0.0
-0.1
0.1
0.3
0.5
Island
0.7
0.9
1.1
Variables

Variables:
Dependent: Proportion of number of
alien species to native species in each
country.
 Independent:

• Island?
• Population Density
• GDP per capita
• Agricultural activity
Computer Minimizes Sei2



Remember, OLS finds coefficients
that minimize sum squared
residuals
Graphical representation
Why is this appropriate?

Can show that this criterion leads to
estimates that are most precise
unbiased estimates.
Dummy Variable

Generally:



Use a “Dummy Variable”. Value = 1 if
country is Island, 0 otherwise.
More generally, if n categories, use n-1
dummies.


Male/Female; Pre-regulation/Post-regulation;
etc..
E.g. if want to distinguish between 6
continents
Problem: Lose “degrees of freedom”.
A Simple Model

A simple linear model looks like this:
Ai  1   2 ISLi  3Pi   4GDPi  5 AGR i  i


Dummy changes intercept (explain).
Interaction dummy variable?

E.g. Invasions of island nations more strongly
affected by agricultural activity.
Translating our Hypotheses

2 Hypotheses




Hypothesis 1: Population: Focus on 3
Hypothesis 2: Island: Focus on 2
“Hypothesis Testing”… forthcoming in course.
Parameter Estimates:
(Intercept)
Island
Pop.dens
GDP
Value
-0.0184
0.0623
0.0010
0.0000
Std.Error t value Pr(>|t|)
0.0859
-0.2141 0.8326
0.0821
0.7588 0.4564
0.0002
6.2330 0.0000
0.0000
3.3249 0.0032
Agr -0.0014 0.0015
-0.9039 0.3763
“Goodness of Fit”: R2
“Coefficient of Determination”
 R2=Squared correlation between Y
and OLS prediction of Y
 R2=% of total variation that is
explained by regression, [0,1]
 OLS maximizes R2.
 Adding independent cannot  R2
 Adjusted R2 penalizes for # vars.

Answers



Island nations are more heavily invaded
(.0623)
 Not significant (p=.46)
Population density has impact on
invasions (.001)
 Significant (p=.0000)
R2=.80; about 80% of variation in
dependent variable explained by model.

Also, corr(A,Ahat)=.89